| 1. | The minimal polynomial for ( hence ) is a factor of.
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| 2. | Using the minimal polynomial of the next smallest Pisot Vijayaraghavan number gives,
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| 3. | Minimal polynomials are useful for constructing and analyzing field extensions.
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| 4. | In the second case, it follows that is the minimal polynomial of.
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| 5. | Minimal polynomials are also used to define conjugate elements.
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| 6. | Now we just need to find its minimal polynomial.
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| 7. | Therefore, is irreducible over by, and the minimal polynomial for is of degree.
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| 8. | While the Jordan normal form determines the minimal polynomial, the converse is not true.
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| 9. | The minimal polynomial is often the same as the characteristic polynomial, but not always.
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| 10. | Because has the same complex roots as the minimal polynomial and because it is real it follows that
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